Linguistic Note 10 Cohomology Group / July 29 2007

Linguistic Note

10

Cohomology Group

　　  TANAKA Akio

Group     G

G-additive group     M

Natural number     n

Gⁿ = { σ₁, … , σ_n | σ_i ∈ G }

Cⁿ = ( G, M ) = { φ : Gⁿ → M | φ is map as set }

C^o = ( G, M ) = M

Against Cⁿ = ( G, M )

( φ + ψ ) ( σ ) = φ ( σ ) + φ( σ )       φ , ψ ∈ Cⁿ      σ∈ Cⁿ  

Element of Cⁿ = ( G, M )     n- Cochain

Homomorphism      dⁿ = Cⁿ  ( G, M )  → Cⁿ⁺¹ ( G, M )    n ≥ 0

 d ⁿ⁺¹. dⁿ  = 0

Zⁿ  ( G, M )  = Ker ( dⁿ )   n ≥ 0

Bⁿ  ( G, M )  = Im ( d^n-1 )  n ≥ 1

Element of Zⁿ  ( G, M ) is n-cosylcle.

Element of Bⁿ  ( G, M )  is n-coboundary.

Bⁿ  ( G, M )  ⊂ Zⁿ  ( G, M )

Cohomology group of M is below.

Hⁿ  ( G, M ) = Zⁿ  ( G, M ) / Bⁿ  ( G, M ) 

H⁰  ( G, M ) = Z⁰  ( G, M )

[Note]

Cohomology group may be helpful to the meaning of words and their variations.

[References]

Property of Quantum     Tokyo May 21, 2004

Prague Theory     Tokyo October 2, 2004

Prague Theory Summary and Prospect     Tokyo October 9, 2004

Theoretical Summarization and Problem in Future     Tokyo November 28, 2004

Tokyo July 29 2007

Sekinan Research Field of Language